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STA 103

Homework 4

True/False

For each of the following questions indicate true or false, then fully explain your answer. You may use examples to illustrate your answer.

(I) The normal distribution is an example of a symmetric distribution.

(II) It is not possible for the mean of a Uniform distribution to be negative.

(III) If two random variables are independent, then it is certain that their correlation is 0.

(IV) A correlation of -0.7 would indicate a weaker linear relationship than a correlation of 0.50.

Full Detail

Work out the following problems. Show your work.

1. Suppose that the average salary of residents in California is 51000, with standard deviation 12000, and assume salary is normally distributed.

(a) If someone’s salary is over 51000, what is the probability it is under 65000?

(b) Find the 15^th percentile of the distribution of the salary of residents in California.

(c) How many standard deviations above the mean is the 85^th percentile of the dis-tribution of the salary of residents in California?

(d) Suppose that the salaries of the husband (X1) and wife (X2) in a household have the same mean, 51000, and same standard deviation, 12000, and their correlation between the salaries is 0.85. Find the variance of the total household salary for the husband and wife.

2. Suppose that average number of times an instructor gets woken up by their cat in 8 hours is 2.3, and assume that a Poisson distribution is appropriate.

(a) Find the probability that the instructor is woken up at most once in 8 hours.

(b) If it is known that the instructor has been woken up at least once, what is the probability they are woken up exactly three times (in the 8 hour period).

(c) Find the average and the standard deviation of this Poisson distribution.

(d) Now suppose an Exponential distribution is appropriate for the time in hours until the instructor is woken up by their cat. I.e, let Y = time in hours until instructor is woken up by their cat. Find E[Y²].

3. Consider the following joint distribution between two discrete random variables X and Y.

Further, assume 

(a) Find E[X] and E[Y].

(b) Determine if X and Y have a positive or negative linear relationship.

(c) Find P(X = 5|Y = 0).

(d) If W = 2X² + 15Y + 2, find E[W].Scratch Work